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ON THE SET OF BETTI ELEMENTS OF A PUISEUX MONOID

Part of: General commutative ring theory Semigroups Arithmetic rings and other special rings

Published online by Cambridge University Press:  20 May 2024

SCOTT T. CHAPMAN Open the ORCID record for SCOTT T. CHAPMAN [Opens in a new window] ,
JOSHUA JANG Open the ORCID record for JOSHUA JANG [Opens in a new window] ,
JASON MAO Open the ORCID record for JASON MAO [Opens in a new window]  and
SKYLER MAO Open the ORCID record for SKYLER MAO [Opens in a new window]
SCOTT T. CHAPMAN*
Affiliation:
Department of Mathematics, Sam Houston State University, Huntsville, TX 77341, USA
JOSHUA JANG
Affiliation:
Oxford Academy, Cypress, CA 90630, USA e-mail: joshdream01@gmail.com
JASON MAO
Affiliation:
The Academy for Mathematics, Science and Engineering, Rockaway, NJ 07866, USA e-mail: jmao142857@gmail.com
SKYLER MAO
Affiliation:
Saratoga School, Saratoga, CA 95070, USA e-mail: skylermao@gmail.com
*
e-mail: scott.chapman@shsu.edu
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Abstract

Let M be a Puiseux monoid, that is, a monoid consisting of nonnegative rationals (under standard addition). In this paper, we study factorisations in atomic Puiseux monoids through the lens of their associated Betti graphs. The Betti graph of $b \in M$ is the graph whose vertices are the factorisations of b with edges between factorisations that share at least one atom. If the Betti graph associated to b is disconnected, then we call b a Betti element of M. We explicitly compute the set of Betti elements for a large class of Puiseux monoids (the atomisations of certain infinite sequences of rationals). The process of atomisation is quite useful in studying the arithmetic of Puiseux monoids, and it has been actively considered in recent literature. This leads to an argument that for every positive integer n, there exists an atomic Puiseux monoid with exactly n Betti elements.

Keywords

nonunique factorisationsPuiseux monoidatomicityBetti graphBetti elementatomisation

MSC classification

Primary: 13F15: Rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
Secondary: 13A05: Divisibility; factorizations 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations 20M13: Arithmetic theory of monoids

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 1 , February 2025 , pp. 80 - 92
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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